Bessel Processes, Schramm–Loewner Evolution, and the Dyson Model

The purpose of this book is to introduce two recent topics in mathematical physics and probability theory: the Schramm–Loewner evolution (SLE) and interacting particle systems related to random matrix theory. A typical example of the latter systems is Dyson's Brownian motion (BM) model. The SLE and Dyson's BM model may be considered as "children" of the Bessel process with parameter D , BES( D ), and the SLE and Dyson's BM model as "grandchildren" of BM. In Chap. 1 the parenthood of BM in diffusion processes is clarified and BES( D ) is defined for any D  ≥ 1. Dependence of the BES( D ) path on its initial value is represented by the Bessel flow. In Chap. 2 SLE is introduced as a complexification of BES( D ). Rich mathematics and physics involved in SLE are due to the nontrivial dependence of the Bessel flow on D . From a result for the Bessel flow, Cardy's formula in Carleson's form is derived for SLE. In Chap. 3 Dyson's BM model with parameter β is introduced as a multivariate extension of BES( D ) with the relation D = β + 1. The book concentrates on the case where β = 2 and calls this case simply the Dyson model.The Dyson model inherits the two aspects of BES(3); hence it has very strong solvability. That is, the process is proved to be determinantal in the sense that all spatio-temporal correlation functions are given by determinants, and all of them are controlled by a single function called the correlation kernel. From the determinantal structure of the Dyson model, the Tracy–Widom distribution is derived.